On Consistent Estimators in Linear and Bilinear Multivariate Errors-In-Variables Models

We consider three multivariate regression models related to the TLS problem. The errors are allowed to have unequal variances. For the model AX = B, the elementwise-weighted TLS estimator is considered. The matrix [A B] is observed with errors and has independent rows, but the errors in a row are correlated. In addition, the corresponding error covariance matrices may differ from row to row and some of the columns are allowed to be error-free. We give mild conditions for weak consistency of the estimator, when the number of rows in A increases. We derive the objective function for the estimator and propose an iterative procedure to compute the solution. In a bilinear model AXB = C, where the data A, B, C are perturbed by errors, an adjusted least squares estimator is considered, which is consistent, i.e. converges to X, as the number m of rows in A and the number q of columns in B increase. A similar approach is applied in a related model, arising in motion analysis. The model is v T Fu = 0, where the vectors u and v are homogeneous coordinates of the projections of the same rigid object point in two images, and F is a rank deficient matrix. Each pair (u, v) is observed with measurement errors. We construct a consistent estimator of F in three steps: a) estimate the measurement error variance, b) construct a preliminary matrix estimate, and c) project that estimate on the subspace of singular matrices. A simulation study illustrates the theoretical results.